Question

If a non•homogeneous system of equations has one dependent row or equation, then (select all that apply): i) the RREF of the augmented matrix will have a row of zeros ii) there will be infinitely many solutions iii) the complementary homogeneous solution will have many solutions iv) the coefficient matrix will not have an inverse v) none of the above

Answer 1

If a consistent non-homogeneous system of equations has an equation which is a scalar multiple of another equation of a linear combination of some other equations in the system, then the system is dependent.

Since nothing has been stated about the size of the matrix A, we presume that A is ac m x n matrix with m ≠ n.

Then the following are correct/true:

i) the RREF of the augmented matrix will have a row of zeros

iv) the coefficient matrix will not have an inverse

As regards ii), we know that if m > n and if there are n non-zero rows in the RREF of A, then there is a unique solution . Hence, ii) may not be correct/true:

As regards iv), we know that if m > n and if there are n non-zero rows in the RREF of A, then the coefficient matrix will have an inverse. Hence, iv) may not be correct/true: